General 1st derivative approximation (obtained by...
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1
General 1st derivative approximation (obtained by Lagrange interpolation)
The interpolation nodes are given as:
…
By Lagrange Interpolation Theorem (Thm 3.3):
∑
(1)
Take 1st derivative for Eq. (1):
∑
( ( ( ))
)
( ( )
) ( )
Set , with being x-coordinate of one of interpolation nodes. .
( ) ∑ ( )
( )
∏
---------- (N+1)-point formula to approximate .
The error of (N+1)-point formula is ( )
∏
.
2
Example. The three-point formula with error to approximate .
Let interpolation nodes be , and .
( ) [
] [
] [
]
( )
∏
Mostly used three-point formula (see Figure 1)
Let , and be equally spaced and the grid spacing be .
Thus and
1.
[ ]
( ) (three-point
endpoint formula)
2.
[ ]
( ) (three-point midpoint
formula)
3.
[ ]
( ) (three-point
endpoint formula)
Mostly used five-point formula
1. Five-point midpoint formula
Figure 1. Schematic for three-point formula